Alternatively, a reduced order model (ROM) has been developed to decrease computational cost while maintaining an acceptable accuracy, i.e., depending on applications, [1]}, [2]}. Recently, the ROM methodology has been applied to a range of parameterized problems (i.e., to repeated evaluations of a problem) depending on a set of parameters (\(\mathbf {\mu }\) ), which could correspond to physical properties, geometric characteristics, or boundary conditions [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}. It is generally composed of two stages, (1) the offline and (2) online stages [12]}. The offline stage begins with initializing a set of input parameters, which we call a training set. Then the FOM is solved corresponding to each member in the training set (in the following, we will refer to the corresponding solutions as snapshots). Data compression techniques, which could be linear or nonlinear compression, are then used to compress FOM snapshots to produce basis functions that span a reduced (linear of nonlinear) space of very low dimensionality but still guarantee accurate reproduction of the snapshots [14]}, [15]}, [16]}. The ROM can then be solved during the online stage for any new value of \(\mathbf {\mu }\)  by seeking an approximated solution in the reduced space.
